Duffing equation (chaos)

This simulation explores the Duffing equation, which reads (in dimensionless variables) as follows:
x'' + 2 γ x' - x (1-x²) = f cos ωt
where each ' denotes a time derivative.

You can select below the parameters γ and f, as well as the initial conditions for the elongation x and the velocity v = x' for the two solutions that are simultaneously computed. Initial conditions can also be selected by moving with the mouse the point on the display Phase space.
The unit time is 1/ ω (so that ω = 1).
For information on other elements, put over them the mouse pointer to get a tooltip.

Activities

Set γ = 0 and f = 0 and solve the system for different initial conditions to see non-linear periodic oscillations.
With the default settings (press Reset to recover them) you can see how the initially small distance between solutions grows in this non-chaotic case
Set γ = 0.1 and f = 0.3 and two very close initial conditions: say x₁ = 1.5, x₂ = 0.500001, v₁ = v₂ = 0. You will see the very definition of deterministic chaos: sensitive dependence on initial conditions.
By changing f, would you be able to find the onset of chaos for a given γ?
The sensitive dependence on initial conditions is also shown in the example Duffing5 of Dynamics Solver.

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.